How should we pool effect sizes in meta-analysis?
2024-07-30
To estimate the true effect size from multiple studies:
Study weight \(w_{k}\) is inversely related to \(SE^2\)
\[ w_{k}=\frac{1}{SE^{2}_{k}} \]
The pooled estimate \(\widehat{\theta}\) is the weighted average:
\[ \widehat{\theta} = \frac{\Sigma_{k=1}^{K} \widehat{\theta}_{k} w_{k}}{\Sigma_{k=1}^{K} w_{k}} \]
A study’s effect size is an estimate of the true study effect size plus sampling error \[ \widehat{\theta}_{k}=\theta_{k} +\epsilon_{k} \]
The true effect size of study k is drawn from a distribution of effect sizes with mean \(\mu\), with error \(\zeta_{k}\).
\[ {\theta}_{k}=\mu+\zeta_{k} \]
\[ w^{*}_{k}=\frac{1}{s^{2}_{k} + \tau^2} \]
And then we estimate the overall mean effect size using the same weighted average as before.
Only choose the FEM if you have clear reason to (rare)
For most scenarios, use the REM.
A simple way to determine whether the REM is appropriate is to test whether the confidenece interval for \(\tau^2\) includes zero or not.